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Riemann Sums and Definite IntegralsB

Riemann Sums and Definite IntegralsB - Riemann Sums and...

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Riemann Sums and Riemann Sums and Definite Integrals Definite Integrals Section 4.3
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Defn. of a Riemann Sum Defn. of a Riemann Sum Let f be defined on the closed interval [a, b], and let be a partition of [a, b] given by a = x 0 < x 1 < x 2 < . . .< x n-1 < x n = b, Where is the length of the i th subinterval. If c i is any point in the i th subinterval, then the sum i x
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x i-1 < c i < x i is called a Riemann sum of f for the partition . c i = a + i 1 ( ) n i i i f c x = x
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The width of the largest subinterval of a partition is the norm of the partition and is denoted by . If every subinterval is of equal length, the partition is regular and the norm is denoted by b a x n - ∆ = ∆ =
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Defn. of a Definite Integral Defn. of a Definite Integral If f is defined on the closed interval [a, b] and the limit exists, then f is integrable on [a, b], and the limit is denoted by 0 1 lim ( ) n i i i f c x ∆ → = 0 1 lim ( ) ( ) b n i i i a f c x f x dx ∆ → = =
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Thm. 4.4 Continuity Implies Thm. 4.4 Continuity Implies Integrability Integrability If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].
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Thm. 4.5
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